\section{Efficient Insurer Portfolio Selections Are Random Samples} 
\label{sec:EfficientInsurerPortfolioSelectionsAreRandomSamples} 

We know that insurer Population Loss Ratio Estimates are uncertain because policyholders' health experiences vary. Very high insurer Population Loss Ratio Estimates occur when one, or more, policyholders produce high individual Population Loss Ratio Estimates. Very low Population Loss Ratio Estimates occur when very few policyholders incur high claims costs. But ``perfect'' years are years in which insurers' Population Loss Ratio Estimates are exactly equal to the Population Loss Ratio. Perfectly efficient insurers have Population Loss Ratio Estimates that are exactly equal to the population loss ratio, not Population Loss Ratio Estimates equal to 0.0000. 

Perfectly efficient insurers become perfectly efficient by issuing infinitely many policies, and because $\sigma_{e_{\infty}}$ = 0.0000, their claims costs equal their Expected Claims Costs. Inefficient insurers have high, or low, Population Loss Ratio Estimates but their ``inefficiency'' is due to their high levels of Population Loss Ratio Estimate variation (i.e. their large standard errors). 

Insurance premiums vary with policyholder risk characteristics and in efficient insurance markets, no insurer can systematically select lower risk policyholders. If this happens high risk policyholders may go uninsured, the most selective insurers will receive excessive premiums, and the least selective insurers, with too many high risk policyholders, will receive inadequate premiums. \emph{In efficient insurance markets, insurers randomly select policyholders.} 

Efficient insurance markets are stochastic processes subject to the Central Limit Theorem (CLT) (Hogg and Craig, 1978). Population Loss Ratio Estimates are ``averages'' calculated from large, randomly selected portfolios of policyholders, and Population Loss Ratio Estimate Cumulative Distribution Functions are normally distributed. I can analyze how portfolio size affects insurers' Population Loss Ratio Estimate variability and operating results by specifying the Population Loss Ratio and either the standard error for a single, Paradigm Insurer ($PI$), or the standard deviation for an individual policyholder. Once these are specified, the CLT specifies all other insurer's standard errors and Population Loss Ratio Estimate Distribution Functions as a function of insurer portfolio size. I then calculate probabilities of specific levels of Population Loss Ratio Estimates and use these to quantify the likely operating results for insurers of any size.

